32 PROCEEDINGS OF THE AMERICAN ACADEMY. 



presenfrcase tin- twisted cubic is degenerate, consisting of D, D ', and G. 

 Let 11- pass a quadric through eight points, three on 1), three on B', one 

 oa ff, and one on any generator .1. the last two points not being on 



l> or D' ; then D, D\ G, and A will all lie mi the quadric and count for 

 7 lines in the intersection of the scroll and quadric, and therefore the 

 quadric cuts out one more generator; the quadric passes through eight 

 fixed points and we can make it pass through an arbitrary ninth point, so 

 it' we vary this ninth point continuously the (juadric will cut out. in suc- 

 cession, each generator of the scroll. .Now < "'" meets the quadric in 2 a 

 points, of which a fixed number lie on D, D', G, and A, and therefore 

 the same Dumber of points of C", say a points, lie on each generator. 

 It is evident that there must be a points on A, for if any other generator 

 be chosen, through which the quadric is always to pass, then there is the 

 same number of points, a, on A, as on each of the other generators. 

 Since we can pass a plane through J) and two generators, there are 

 a — 2a points of a a on D, and, similarly, there are a — 2 a points of 

 a a on D'. A plane through D and G meets the scroll in these two lines 

 only, and then- are, therefore, 2u points of a a on G, as is otherwise evi- 

 dent from the fact that G counts for two generators. Since a twisted 



curve of order a cannot have a points on any line, 2 a < a or a < for 



every twisted curve on the scroll. 



3. Phtne Curves. — Each double director is met once by any gener- 

 ator and is therefore a 2,. No generator can meet G\ for. BUppose a 

 generator .1 does meet it; then A meets either Dor /)' in a point differ- 

 ent from that in which G meet- it. and therefore the plane through G 

 and A contains also I) or 27, making the order of the complete intersec- 

 tion of the plane and scroll as great a- 5, which is impossible. G is 



therefore a 2,,. Then any plane through <•■ that doe- not contain I) or //, 

 meet- the scroll in a proper conic that doe- not meet either double 

 director, since the section has only a double point on each double director ; 

 -ince each <_'<'iierator meets the plane mice and doe- not meet (,\ each 

 conic is a 2,, and the COnicS and D and //are theonh cur\es on the 



u 



BCroll for which u = -. A plane through one and onlv one generator 



2 i . 



cuts out a plate cubic, a -5,. having a double point on G and passing once 

 through the two points where the generator in the plane me< ts 1 > and I > \ 

 the generator meet-, the cubic again where the plane i- tangeul to the 



scroll. A plain- that do.- not contain a line of the scroll cuts out a plane 



quartic, a l,. having three double points, one on G and one on each 



